Solve {l}{31x-16y-7z=80}{-16x+27y-4z=0}{-7x-4y+31z+24y=0}

$$\left\{ \begin{array} { l } { 31 x - 16 y - 7 z = 80 } \\ { - 16 x + 27 y - 4 z = 0 } \\ { - 7 x - 4 y + 31 z + 24 y = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

$$y=2$$

$$z=-\frac{1}{2}=-0.5$$

Short Steps Using Substitution

Solve 31x-16y-7z=80 for x.

$$x=\frac{16}{31}y+\frac{7}{31}z+\frac{80}{31}$$

Substitute \frac{16}{31}y+\frac{7}{31}z+\frac{80}{31} for x in the second and third equation.

$$-16\left(\frac{16}{31}y+\frac{7}{31}z+\frac{80}{31}\right)+27y-4z=0 -7\left(\frac{16}{31}y+\frac{7}{31}z+\frac{80}{31}\right)-4y+31z+24y=0$$

Solve these equations for y and z respectively.

$$y=\frac{1280}{581}+\frac{236}{581}z z=\frac{35}{57}-\frac{127}{228}y$$

Substitute \frac{1280}{581}+\frac{236}{581}z for y in the equation z=\frac{35}{57}-\frac{127}{228}y.

$$z=\frac{35}{57}-\frac{127}{228}\left(\frac{1280}{581}+\frac{236}{581}z\right)$$

Solve z=\frac{35}{57}-\frac{127}{228}\left(\frac{1280}{581}+\frac{236}{581}z\right) for z.

$$z=-\frac{1}{2}$$

Substitute -\frac{1}{2} for z in the equation y=\frac{1280}{581}+\frac{236}{581}z.

$$y=\frac{1280}{581}+\frac{236}{581}\left(-\frac{1}{2}\right)$$

Calculate y from y=\frac{1280}{581}+\frac{236}{581}\left(-\frac{1}{2}\right).

$$y=2$$

Substitute 2 for y and -\frac{1}{2} for z in the equation x=\frac{16}{31}y+\frac{7}{31}z+\frac{80}{31}.

$$x=\frac{16}{31}\times 2+\frac{7}{31}\left(-\frac{1}{2}\right)+\frac{80}{31}$$

Calculate x from x=\frac{16}{31}\times 2+\frac{7}{31}\left(-\frac{1}{2}\right)+\frac{80}{31}.

$$x=\frac{7}{2}$$

The system is now solved.

$$x=\frac{7}{2} y=2 z=-\frac{1}{2}$$